Multiplicity of solutions for fractional q(.)-Laplacian equations

2021/03/23 by Rahmoune, Abita, Biccari, Umberto
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.12600

Abstract

In this paper, we deal with the following elliptic type problem \begincases (-Δ)_q(.)^s(.)u + λVu = α\vert u\vert^p(.)-2u+β\vert u\vert^k(.)-2u amp; in Ω,
u =0 amp; in ℝⁿ\backslash Ω, \endcases where q(.):Ω× Ω→ ℝ is a measurable function and s(.):ℝⁿ× ℝⁿ→ (0,1) is a continuous function, n>q(x,y)s(x,y) for all (x,y)∈ Ω× Ω, (-Δ)_q(.)^s(.) is the variable-order fractional Laplace operator, and V is a positive continuous potential. Using the mountain pass category theorem and Ekeland's variational principle, we obtain the existence of a least two different solutions for all λ>0. Besides, we prove that these solutions converge to two of the infinitely many solutions of a limit problem as λ→ +∞ .

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